THE IMPACT OF UNMODELED TIME SERIES PROCESSES IN WITHIN-SUBJECT RESIDUAL STRUCTURE IN CONDITIONAL LATENT GROWTH MODELING: A MONTE CARLO STUDY

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1 THE IMPACT OF UNMODELED TIME SERIES PROCESSES IN WITHIN-SUBJECT RESIDUAL STRUCTURE IN CONDITIONAL LATENT GROWTH MODELING: A MONTE CARLO STUDY By YUYING SHI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 009 1

2 009 Yuying Shi

3 To my family in China 3

4 ACKNOWLEDGMENTS First, I would like to express my sincere gratitude to my two professors, Dr. Leite and Dr. Algina. Dr. Leite is my advisor and the chair of the committee. He is always available to answer my questions and encourages me all through the four years study. I greatly appreciate his instructive advice, patience and valuable suggestions on my dissertation. His help is indispensable for my first publication and the dissertation. Dr. Algina is another professor that I should show my deepest gratitude to. A respectable, responsible and resourceful scholar, he has provided me with valuable guidance in every stage of my doctoral study. With his expertise in many aspects, he enlightens me not only in the dissertation but also in my future research. My great gratitude also goes to other two committee members, Dr. Miller and Dr. Huang. Dr. Miller is an exceptional professor. His keen insights, breadth of knowledge, flexibility and approachable manner have been an invaluable source to me. My thanks also go to Dr. Huang for his valuable suggestions and generous encouragement through this study. I would like to thank many friends at University of Florida for making my academic life more enjoyable. Special thanks should go to my two best friends, Hong and Feiqi. They are always there listening to me and supporting me. My true words are beyond the gratitude to my beloved family for their unconditional love, understanding and continuous care. They are the best gift that I can ever receive from the God. 4

5 TABLE OF CONTENTS ACKNOWLEDGMENTS... 4 LIST OF TABLES... 8 LIST OF FIGURES ABSTRACT... 1 CHAPTER 1 INTRODUCTION LITERATURE REVIEW page Latent Growth Model Unconditional Latent Growth Model... 0 Conditional Latent Growth Model... 7 Latent growth model with a time-invariant covariate... 7 Latent growth model with a time-varying covariate Latent growth model with a parallel process Assumptions of Growth Modeling Within-person residual covariance structure Measurement time and missing data Functional form of development Comparisons with Other Methods Stationary Time Series Model Autoregressive (AR) Model Moving Average (MA) Model Autoregressive Moving Average (ARMA) Model Modeling Time Series in the Error Structure in Longitudinal Data Analysis... 5 Studies on the Impact of Misspecifying the Within-Person Error Structure Significance of This Study Research Questions METHOD Design Factors Number of Measurement Times Sample Size Time Series Parameters Time Coding Population Values

6 Within-Person Residual Variance σ... 6 Parameter µ α and µ B in Between-Person Equation Residual Variance of Level Equation (i.e.,,), Residual Variance of Shape Equation (i.e., ), and Covariance of Level and Shape Residuals (i.e., ) Mean and Variance of Time Invariant Covariate Parameters of Time Varying Covariate Effect of Time Invariant Predictor on Latent Level and Latent Shape in Growth Predictor Model (i.e., γ α and γ β in Equation -18) Effect of the Time Varying Predictor Variable on the Outcome Variable in LGM with a time varying Covariant (i.e., γ t in Equation -7) Effect of the Intercept and Slope of the Predictor on the Intercept and Slope of the Outcome Variable in LGM with a parallel process Model Summary of Population Values LGM with a Time Invariant Covariate LGM with a Time Varying Covariate LGM with a parallel process Summary of Conditions Data Generation Data Analysis RESULTS Convergence Rate and Non-Positive Definite Covariance Matrix Occurrence Rate Fixed Parameter Estimates LGM with a Time Invariant Covariate LGM with a Time Varying Covariate LGM with a parallel process Standard Error of the Fixed Parameter Estimates LGM with a Time Invariant Covariate LGM with a Time Varying Covariate LGM with a parallel process Summary of the Results for the Fixed Parameter Estimates together with Standard Error Estimates Variance Component Parameter Estimates AR (1) Within-Person Residual Covariance Matrix MA (1) Within-Person Residual Covariance Matrix ARMA (1, 1) Within-Person Residual Covariance Matrix Summary of the Results for Variance Component Parameter Estimates Standard Error Estimates of Variance Components AR (1) Within-Person Residual Covariance Matrix MA (1) Within-Person Residual Covariance Matrix ARMA (1, 1) Within-Person Residual Covariance Matrix Summary of Standard Error Estimates of the Variance Components Chi-Square GOF Test and GOF Indexes

7 GOF Test AR (1) within-person residual covariance matrix MA (1) within-person residual covariance matrix ARMA (1, 1) within-person residual covariance matrix Summary of results for GOF test TLI and CFI AR (1) within-person residual covariance matrix MA (1) within-person residual covariance matrix ARMA (1, 1) within-person residual covariance matrix Summary of results for CFI and TLI RMSEA and SRMR AR (1) within-person residual covariance matrix MA (1) within-person residual covariance matrix ARMA (1, 1) Within-Person Residual Covariance Matrix Summary of results of SRMR and RMSEA Summary of GOF test and GOF indexes DISCUSSION AND CONCLUSION General Conclusions and Discussions Summary of Impact of Each Factor Impact of Analysis Model Type Impact of Time Series Parameter Impact of Sample Size Impact of Length of Waves Analytic Results of Variance Components Estimates GOF Test and GOF Indexes Suggestions to Applied Researchers Limitations and Suggestions for Future Research APPENDIX: MPLUS CODE Latent Growth Model with a Time Invariant Covariate with an AR (1) Process Latent Growth Model with a Time Invariant Covariate with an MA (1) Process Latent Growth Model with a Time Invariant Covariate with an ARMA (1, 1) Process LIST OF REFERENCES BIOGRAPHICAL SKETCH

8 LIST OF TABLES Table page 4-1 Convergence rate for all conditions Rate of occurrence of non-positive definite matrix under all conditions Marginal mean relative biases of fixed parameter estimates for LGM with a time invariant covariate Mean relative biases of fixed parameter estimates for LGM with a time varying covariate Mean relative biases of fixed parameter estimates for LGM with a parallel process Marginal mean relative biases of standard error estimates of fixed parameters for LGM with a time invariant covariate Marginal mean relative biases of standard error estimates of fixed parameters for LGM with a time varying covariate Marginal mean relative biases of standard error estimates of fixed parameters for LGM with a parallel process Mean relative biases of estimates for three LGMs with an AR (1) within-person residual covariance matrix Mean relative biases of estimates for three LGMs with an AR (1) within-person residual covariance matrix Mean relative biases of estimates for three LGMs with an AR (1) within-person residual covariance matrix Mean relative biases of estimates for three LGMs with a MA (1) within-person residual covariance matrix Mean relative biases of estimates for three LGMs with a MA (1) within-person residual covariance matrix Mean relative biases of estimates for three LGMs with a MA (1) within-person residual covariance matrix Mean relative biases of estimates for three LGMs with an ARMA (1, 1) withinperson residual covariance matrix

9 4-16 Mean relative biases of estimates for three LGMs with an ARMA (1, 1) withinperson residual covariance matrix Mean relative biases of estimates for three LGMs with an ARMA (1, 1) withinperson residual matrix, collapsing across sample size Mean relative biases of standard error estimates of for three LGMs with an AR (1) within-person residual covariance matrix Mean relative biases of standard error estimates of and for three LGMs with an AR (1) within-person residual covariance matrix Mean relative biases of standard error estimates of variance components for three LGMs with a MA (1) within-person residual covariance matrix Mean relative biases of standard error estimates of for three LGMs with an ARMA (1, 1) within-person residual covariance matrix Mean relative biases of standard error estimates of for three LGMs with an ARMA (1, 1) within-person residual covariance matrix Mean relative biases of standard error estimates of for three LGMs with an ARMA (1, 1) within-person residual covariance matrix Percentage of p value below 0.05 for three LGMs with an AR (1) within-person residual covariance matrix Percentage of p value below 0.05 for three LGMs with a MA (1) within-person residual covariance matrix Percentage of p value below 0.05 for three LGMs with an ARMA (1, 1) withinperson residual covariance matrix Percentage of TLI and CFI statistics that indicated adequate model fit for three LGMs with an AR (1) within-person residual covariance matrix Percentage of TLI and CFI statistics that indicated adequate model fit for three LGMs with a MA (1) within-person residual covariance matrix Percentage of TLI and CFI statistics that indicated adequate model fit for three LGMs with an ARMA (1, 1) within-person residual covariance matrix Percentage of RMSEA and SRMR statistics that indicated adequate model fit for three LGMs with an AR (1) within-person residual covariance matrix

10 4-31 Percentage of RMSEA and SRMR statistics that indicated adequate model fit for three LGMs with a MA (1, 1) within-person residual covariance matrix Percentage of RMSEA and SRMR statistics that indicated adequate model fit for three LGMs with an ARMA (1, 1) within-person residual covariance matrix Biases of obtained with three data sets for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix Biases of obtained with three data sets for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix Biases of obtained with three data sets for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix Biases of standard error estimates of obtained with three data sets for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix Biases of standard error estimates of obtained with three data sets for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix Biases of standard error estimates of obtained with three data sets for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix The frequency table for the standard error estimates of for LGM with a parallel process with an ARMA (1, 1) within-person residual covariance matrix Biases of standard error of estimates obtained with and without imposing starting values for LGM with a time invariant covariate with an AR (1) withinperson covariance matrix The frequency table for the standard error estimates of under LGM with a time invariant covariate with an AR (1) within-person covariance matrix Biases of standard error estimates of obtained with two data sets for LGM with a time invariant covariate with an AR (1) within-person residual covariance matrix

11 LIST OF FIGURES Figure page -1 Unconditional latent growth model Latent growth model with a time invariant covariate Latent growth model with a time varying covariate Latent growth model with a parallel process

12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE IMPACT OF UNMODELED TIME SERIES PROCESSES IN WITHIN-SUBJECT RESIDUAL STRUCTURE IN CONDITIONAL LATENT GROWTH MODELING: A MONTE CARLO STUDY Chair: Walter Leite Major: Research and Evaluation Methodology By Yuying Shi August 009 As latent growth modeling is a popular method for analyzing longitudinal data, it is worthy of methodologists attention to investigate the consequences of model misspecification. This study specifically investigated the impact of unmodeled time series processes in the withinperson residual covariance structure on the parameter estimates and standard error estimates, as well as on the chi-square goodness of fit test and some commonly used fit indexes. It was found that when the analysis model failed to include any type of time series process, all the fixed parameter estimates, together with their standard error estimates, were not affected. The variance components estimates were biased to different degrees under some conditions, depending on the type of within-person residual covariance structure. The standard error estimates of these variance components were not affected by model misspecification. Based on the results, it is recommended that applied researchers consider alternative covariance structures. It was also found that when the within-person residual covariance structure is an AR (1) or a MA (1) process, the chi-square goodness of fit test and RMSEA can be used for model selection under many conditions. However TLI could be used to detect model misspecification for only one condition, while CFI and SRMR were not reliable in model differentiation. When 1

13 the within-person residual covariance structure was an ARMA (1, 1) process, only RMSEA could be used for model selection under certain conditions. 13

14 CHAPTER 1 INTRODUCTION Longitudinal data, also called panel data, have been frequently encountered in social and behavioral sciences. A longitudinal data set contains observations of a number of subjects (individuals, firms, countries, etc.) measured over two or more time periods. For example, in educational research, a typical longitudinal data set contains the academic scores of a number of students measured at different time periods. Such data sets provide a large number of observations for a single individual subject and therefore greatly increase the degree of freedom in model estimation. The most important advantage of a longitudinal data set is that it allows researchers to investigate questions that could not be addressed by using just cross sectional data. For example, with a typical education longitudinal data set, researchers can measure the change or growth of the academic performance among students within a specified time period and can identify what factors affect their growth during this period. Such growth investigation could not be implemented with the cross-sectional data. The popularity of studying change has been reflected in the availability of many largescale national longitudinal data in social science. In the education field, widely used longitudinal data sets include the National Education Longitudinal Study (NELS), High School and Beyond (HSB), Early Childhood Longitudinal Study (ECLS), and National Longitudinal Study of Youth (NLSY). In economics fields, some prominent longitudinal data set such as the National Longitudinal Surveys of Labor Market Experience (NLS) and the University of Michigan s Panel Study of Income Dynamics (PSID) have been widely analyzed. Accompanying the widely available data sets, a variety of methods for analyzing longitudinal data have emerged. The commonly used methods include analysis of variance (ANOVA), multivariate analysis of variance (MANOVA), hierarchical linear modeling (HLM), 14

15 generalized linear model (GLM), fixed effects model, random effects model, and latent growth modeling. Each method has its own advantages and limitations. Their applicability depends on the actual research design and research questions that are of interest. Among all these models, latent growth model (LGM), also called latent curve model, growth curve model, emerged relatively recently but gained increasingly popularity. Moreover, with the recent development of more complex LGM, such as the mixed effect LGM, multilevel LGM, multivariate LGM, latent growth modeling becomes a powerful tool in various situations involving longitudinal data analysis. To see how popular the latent growth modeling method is in social science research, a search in Academic Search Premier, Business Source Premier, EconLit, Professional Development Collection, Psychology and Behavioral Sciences Collection, PsycINFO, Psychology and Behavioral Sciences Collection, Sociological Collection using the key word latent growth model in the peer review articles ranging from January 000 to December 008 resulted in 931 articles, which is sound proof of the popularity of this method. LGM is composed of the trajectory equation (also called the within-subject or withinperson equation) and the level and shape equation (also called the between-subject or betweenperson equation). The trajectory equation describes the growth trend of each individual. It contains an error term that captures all the unobserved characteristics for a single individual. The level and shape equation describes the latent level and latent shape respectively for all the individuals. In both the level and shape equation, a between-person error term is included to model the variation of growth level or growth trend between people. The error in the within-person equation describes the difference between the value of observed outcome variable and the value predicted by the trajectory equation. It captures all the unmeasured factors for an individual, such as his/her ability, education level, health status or an 15

16 event that might affect this person s growth. LGM, compared with traditional methods, such as ANOVA, MANOVA, gives substantial flexibility in specifying the within-person residual covariance structure. However, most applied researchers typically assume the within-person residuals are multivariate normally distributed with mean of zero and constant variance. That is, each individual has equal variance across time periods and the errors are independent across time. Under this assumption, the correlation of observed scores at any two time points is due solely to the presence of between-person variation. This simplification brings some concerns. First, some of the important aspects of change might be captured by the within-person residuals (Biesanz, West, & Kwok, 003; Hedeker & Mermelstein, 007). For example, Hedeker and Mermelstein (007) showed that mood change in the smokers could be reflected in the withinperson residual covariance structure rather than in the average change. Second, as mentioned before, anything unmeasured but specific to an individual could be reflected in the within-person error term. If these characteristics remain approximately constant over the sample period, then the independence assumption of the within-person residual seems reasonable. If these characteristics vary over the sample period, the assumption is less realistic. It is not an unreasonable conjecture that some of the events might affect the individual over time. Consider evaluating the reading ability of kids in elementary school. The reading performance of a child might be increasing at a relatively constant rate, but individual observations might deviate from this general trend due to a number of factors in the individual s growth period (e.g., a health problem or a family crisis). Previous studies have shown that correlated measurement errors often exist in longitudinal data (e.g., Fitzmaurice, Laired, & Ware, 004; Joreskog, 1979; Marsh, 1993; Rogosa, 1979; Sivo, 1997; Sivo & Willson, 1998). Therefore, the simple uncorrelated within-person error structure can not fully represent the data characteristics. A variety of more 16

17 complex within-person residual structures have been identified, such as Toeplitz or moving average, autoregressive, compound symmetry and etc. (e.g. Goldstein, 1995; Wolfinger, 1993). Third, when the within-person residual covariance structure is misspecified, the parameter estimate might be affected and the inference based on these estimates might be inaccurate. Various studies have been conducted on the impact of assumption violations in the within-person residual covariance structure on model parameters estimates (e.g., Yuan & Bentler, 004; Ferron, Dailey, & Yi, 00; Singer & Willett, 003). See Chapter for a presentation of results. This study considers three time series within-person error structures: first-order autoregressive (AR) process, first-order moving average (MA) process and first order autoregressive and moving average (ARMA) process. The three times series are commonly encountered in time series analysis. The three kinds of residual covariance structures, although have been well discussed in fields like the econometrics, are relatively unpopular in education field. LGM could be classified as unconditional LGM and conditional LGM. The two types of models differ in whether covariates are added in the model. In unconditional LGM, no time varying or time invariant covariate is added in the model, whereas in conditional LGM at least one covariate is included. Within applications of LGM, most applied research is conducted within the framework of conditional LGM, because conditional LGM enables researchers to include predictors and thus to capture the relationship between individual characteristics and growth parameter. However, most previous studies on model misspecification were conducted within the framework of unconditional LGM (e.g. Sivo, Fan & Witta, 005; You, 006). Although these studies with unconditional LGM shed some light on the possible consequence of model misspecification, whether those results could be generalized to conditional LGM is 17

18 unknown. In unconditional LGM, the parameters of interest are mean, variance and covariance of the latent intercept and latent shape. With the inclusion of time varying and time invariant predictors, conditional LGM involves more parameters estimates, for instance, the direct effect of the predictor on latent factors. Therefore, the impact of model misspecification might be different from those occur in unconditional LGM. Moreover, even though the AR process has been well discussed in the context of LGM, up to now, very few studies include a systematic discussion of AR, MA and ARMA at the same time. Given their popularity and importance in time series analysis, they deserve a systematic application in longitudinal data analysis. Furthermore, no studies have been conducted to investigate the consequence of three unmodeled time series processes on conditional LGM. The three conditional LGMs investigated in this study are: LGM with a time invariant covariate, LGM with a time varying covariate and LGM with a parallel process. These three types of LGMs are representatives of the typical conditional LGMs in applied research. They describe the standard way of including predictors and are commonly used. The goal of this study is to investigate the impact of unmodeled time series processes in latent growth modeling through a Monte Carlo simulation study. To be specific, this study aims to evaluate how the model parameters estimates and standard errors, as well as GOF test and fit indices are affected when the within-person residual covariance structure demonstrates a time series process but the researchers fail to model these processes. This is an area less investigated in LGM. This study is believed to be an important contribution in empirically examining the impact of model misspecification and could provide researchers with better understanding of the consequence of assumption violation in growth modeling and provide useful information for handling these problems. 18

19 CHAPTER LITERATURE REVIEW This chapter is composed of six parts. The first part introduces the unconditional LGM and three types of conditional LGMs, with a general picture presented in the beginning of the first part, and the basic assumptions in LGM introduced at the end of the first part. Then the comparison between LGM with other methods is presented in the second part. The time series models are introduced in the third part, together with studies regarding modeling time series in the error structure in longitudinal data analysis. Followed in the fifth part are previous studies on the impact of model misspecifications. The sixth part presents the research questions and discusses the importance of this study. Latent Growth Model LGM can describe the individual change in a variety of ways: It can describe the individual initial status and growth trend, which can be linear, quadratic or other functional forms; It can estimate the variability across individuals in both initial level and trajectories, and can provide a means for testing the contribution of other predictors to the initial status and growth trajectories. Latent growth modeling methods accomplish these functions by analyzing not only the covariance structure but also the mean structure of variables. In other words, it can simultaneously estimate the changes in covariances, variances and means. The covariance structure contains information about individual differences while mean structure captures information at the aggregate level. In LGM, there are three important latent factors: level, shape and error, which will be illustrated in the subsequent parts. The analytic interest in LGM is not specifically on the indicators but on the latent factors. Each outcome variable measured at any time is a function of these three latent factors. One of the advantages of LGM is that it allows the level and shape to 19

20 vary across individuals under the assumption that the conceptualization is correct. The level represents the status of individuals in terms of the outcome variable at the measurement time set as a reference. If the first measurement time is taken as reference, the level can also be interpreted as the intercept (Muthén & Khoo, 1998). The level of an individual keeps constant across all measurement times. For different people, the level can be different from the beginning. The shape factor, describes the rate of change across time. When the growth trend is linear, the shape is interpreted as a slope. The errors capture the deviation from the observed variables to the estimators obtained from the trajectory (within-person) model. The errors come from a variety of sources: it could be measurement error (e.g. the error caused by instrument or rater unreliability) or systematic error (e.g. the error due to unobserved variables or model misspecification of functional form). Unconditional Latent Growth Model As described in the introduction, the unconditional latent growth model refers to a model without predictors (See Figure -1). The trajectory equation (within-person equation) for this model is expressed as follows: yit = αi + λβ t i + εit, (-1) where yit is the outcome variable measured for the ith individual at time t. For a simple illustration, data are assumed to be collected in four equally spaced measurement times. All the subsequent introduced formulas would follow the four waves pattern. Therefore, t =1,, 3, 4. Parameter α refers to the level for the i th subject while parameter β is the shape for the i th subject. The α i and i β i are considered latent factors. The parameters i α i and β i are allowed to differ across individuals. The variable ε it is the trajectory equation error of i th individual at time t with E( ε it ) = 0. More about the εit will be discussed later. 0

21 σ ζ ζ α i βi ζ α i 1 ζ β i μ α μ β α i β i λ 1 λ λ t y y i1 i y it ε i1 ε i ε it Figure -1. Unconditional latent growth model Parameter λ t refers to the factor loading of latent shape. The λt is fixed as t 1 across all measurement times. That is, λ t =0, 1,, 3, which means all the measurements are taken at equally spaced time points. If the measurement is not taken at equal intervals, for instance, it is taken at month 1, month, month 3.5, and month 6, the λt are specified as 0,, 3.5, 6. When the loading is fixed to be zero, the time the zero loading represents is called reference point of development. In the above example, month 1 is considered as reference point. In this case, 1

22 parameter λt represents the elapsed time from the reference point to time t. The functional form is linear, which means for equal time periods a given individual is growing by the same amount. The individual level and shape can be decomposed into: α = µ + ζ i α αi B i = µ B + ζ Bi, (-) where µ α and µ B are the mean level and mean shape respectively. The mean level represents the average individual initial status. The mean shape represents the average growth rate across all sampled individuals. A positive µ B indicates that on average individuals grow in the observed variable while a negative µ B indicates a average growth decrease in the observed variable. The parameters ζ αi and ζ Bi are the disturbances of level and shape respectively with mean of zero and variances of and, as well as covariance of. In unconditional models, the variances of these two disturbances (i.e., ζ αi and ζ Bi ) also represent the variance of the level and shape respectively. However, the interpretation is not the same when predictors are included in the level and shape equations. When predictors are included (see the subsequent introduction of conditional LGM), the variances of these two disturbances become residual variances, which are interpreted as the variability leftover in the level and shape factor after controlling the effects of predictors. A higher and indicate that sample subjects are more diverse. In the extreme case when ζ αi and ζ Bi are all zero, there is no variability of level and shape across all people, which means all individuals have the same intercept and slope for their growth trajectories. A non-zero variance ofζ indicates that the sampled individuals differ from each other from the αi beginning of the study. A non-zero variance of ζ Bi indicates that individuals grow at different rates. Hence, adding predictors in the model can help to account for the variability of individual

23 growth (Willet & Keiley, 000). Therefore, the level and shape equation describes the individual difference across the whole sample. The covariance between α i and B i represents the relationship between the level and growth trajectory. The equation - 1 and equation - can be combined to a complete model: = µ + λµ + ζ + λζ + ε. (-3) yit α t β αi t βi it This combined model is also called reduced form equation (Bollen & Curran, 005) in that that the endogenous term α i and β i are replaced by their exogenous predictors and disturbances. The variable y it is a combination of fixed component and random component, where the fixed component refers to the term µ + λµ, and random component refers to the term α t β ζ + λζ + ε. It should be noted that here the random component is heteroscedastic across αi t βi it time due to the effect of λ t, which varies over time. Equation -1 describes a linear trajectory relationship between the measurement time and individual growth change. If we want to extend this linear relationship to the broader class of nonlinear relationship, a simple way is to add higher-order polynomial terms. For example, a quadratic equation becomes: y = α + λβ + λ β + ε, (-4) it i t 1i t i it where λt is simply the squared value of time at measurement time t ; β 1i is the slope for the linear term and βi is the slope for the quadratic term of the curve. The interpretations of other components of the equation remain the same. Similarly, we can incorporate cubic, quartic or other higher-power terms of time in this model. In equation -4, as the function does not describe a linear relationship anymore, the change of y is not the same for equal time passage. For instance, assuming measurement at equal intervals and the reference point is time 1, the change 3

24 of y from time 1 to time is equal to β1 i + βi, but the change of y from time to time 3 is β + 5β. In the function describing linear relationship (equation -1), the change of y between 1i i any two time periods always equals β i. The level and shape equation corresponding to the equation -4 is: α i = µ α + ζαi β = µ + ζ 1i β1 β1i β = µ + ζ i β βi. (-5) Equation -5 is similar to equation - except the addition of the equation for the quadratic slope B i. The B i, similarly as αi and B 1i, is randomly varying across individuals. The structural equation form of the above linear trajectory equations could be expressed employing LISREL format (e.g. Muthén & Khoo, 1998; Singer & Willet, 003, Bollen & Curran 005). The LISREL formula is presented as follows: yi =Λ ηi + εi, (-6) where y i is a T x 1 vector of repeated measures, Λ is a T x m matrix of factor loadings, where m is the number of latent factors, η i is an m x 1 vector of latent factors, and ε i is a T x 1 vector of random errors. The matrix format of each term in equation -6 can be illustrated as follows, assuming four repeated measures: yi1 1 0 εi1 y i 1 1 α i ε i yi =, Λ=, ηi =, εi =. y i3 1 β i ε i3 y 1 3 ε i4 i4 (-7) η i can be expressed as: 4

25 η = µ + ζ, (-8) i η i α i µ α where ηi = β, µ η = i µ, ζ B ζ. α η = ζ B The expression of y can be obtained by combining equation -7 and equation -8: y =Λ µ + ( Λ ζ + ε). (-9) η The model implied variance of the above equation is = ΛΨΛ + Θ (-10) ' ( θ ) ε, where Ψ is the covariance matrix of ζ, and of the residuals of the outcome variable y. Θ ε represents the variance and covariance matrix The elements of Ψ and Θε are: Ψ= βα, (-11) where = var( ζ ), = var( ζ ) and = cov( ζ, ζ ) and (still assuming T = 4) α i β i α β i i Θ ε = σ σ e σ e4 e1 0 σ e 0 0. (-1) When the estimated model fits the data, the following equality holds: = ( θ ), (-13) where is the population covariance matrix of the y s, ( θ ) is the model implied covariance matrix of the ' y s. The elements of are 5

26 σ y σ 1 y1y σ y1y4 σ yy σ 1 y σ yy 4 =, (-14) σ yy... σ y σ 3 yy 3 4 σ y y σ 1 y4y σ y4 The model implied covariance matrix for the observed variables is ( θ ) + λ1 + λ 1 + σe + λλ 1 1 t + ( λ1+ λt) + λ λ + ( λ + λ ) + λ λ + ( λ + λ ) + λλ + ( λ + λ ) + λ + λ + σ 1 1 t t = t 1 t 1 t t et. (-15) Finally, the expected value of the outcome variable equals μ =Λ μ. (-16) y η The model implied mean structure is μ( θ ). When the expected mean μ y is equal to the model implied mean structure μ, the following equation should be obtained, in vector notation: y μ y1 μα + λμ 1 β μ y μα λμ + β =. (-17) μ yt μ α + λμ T β The unconditional latent growth modeling is the simplest form of latent growth modeling. In practice, many researchers fit an unconditional growth model before fitting any type of more sophisticated LGM, such as conditional LGM, multilevel LGM, mixture LGM. The unconditional LGM could be used to establish the correct growth trajectory. Furthermore, the unconditional LGM describe the variability of the level and shape and serve as an assessment of whether adding predictors is justified. In general, the unconditional LGM is the first step in many LGM applied studies. 6

27 Conditional Latent Growth Model As mentioned above, adding predictors in the model can help to account for the variability of individual growth. In many situations, researchers are interested in more complex research questions. The conditional growth model provides a convenient way to test various hypotheses. For instance, if we want to estimate the change of children s math skill by controlling their social economics status (SES), SES can be added as a predictor in the model. LGM allows us to incorporate predictors in the model in extremely flexible ways, which will be illustrated in the subsequent examples. Predictors could be time invariant or time varying. Time invariant predictors refer to variables that are constant across time, such as gender, nationality and ethnicity. Time varying predictors, on the contrary, refer to predictors that change as time passes by, such as students test performance, marital status, individual s ability, and so on. The conditional LGMs that were investigated in this study were LGM with a time invariant predictor, LGM with a time varying covariate and LGM with a parallel process. These are commonly used conditional LGM in applied research. Latent growth model with a time-invariant covariate For a simple illustration, only one predictor measured without error is included (See Figure -). In real situations, more than one predictor can be incorporated into the model. The trajectory equation is still the same as that in unconditional model: yit = αi + λβ t i + εit (-18) The level and shape equation is different from that in unconditional model: α i = µ α + γαxi + ζαi, (-19) B = µ + γ x + ζ i B B i Bi 7

28 where μα and μ B are the mean level and mean slope respectively when predictor x is set to zero.. The parameters ζ αi and ζ Bi are the disturbances of level and shape respectively after controlling the effect of the predictor x. As mentioned before, in unconditional models, the variances of these two disturbances also represent the variance of the level and shape respectively. However, once predictors are included in the level and shape equations, the ζ and ζ Bi can not be simply interpreted as the variance of the level and shape respectively any more. The coefficients γ α1 and γ B1 are the direct effects of x variable on level and shape respectively. αi x i γ α μ α 1 γ μ β β ζ α i α i β i ζ β i λ 1 λ λ t y i1 yi y it ε i1 ε i ε it Figure -. Latent growth model with a time invariant covariate 8

29 waves): The structural equation form of the model is represented as follows (still assuming four yi =Λ ηi + εi, (-0) where y i yi1 y i =, yi3 yi εi1 1 1 α Λ= i ε i, η 1 i = β, εi = i εi3 1 3 ε i4 η = µ +Γ x + ζ, (-1) i η i i α i where ηi = β, i µ µ, α η = µ B γ α1 Γ= γ, β1 ζ αi ζη = ζ βi The combined model is obtained by substituting η i in equation -1 into equation -0: y =Λ ( µ +Γ x ) +Λ ζ + ε. (-) i η i i i The implied mean structure is µ =Λ ( µ +Γ µ ). (-3) y η χ The model implied covariance structure could be derived by using deviation score to simplify the analytical expression of the implied covariance matrix (Bollen and Curran, 005). The deviation score formula is presented as follows: yi y η i i i η χ i χ i i µ = [ Λ ( µ +Γ χ ) +Λ ζ + ε ] [ Λ ( µ +Γ µ )] =Λ( Γ( χ µ ) + ζ ) + ε. (-4) The model implied covariance matrix is 9

30 where ' yy ( θ) yx ( θ) yi µ y yi µ y ( θ ) = E xy ( θ) xx ( θ ) = x i µ x x i µ x xx ' ' E ( yi µ y)( yi µ y) E ( yi µ y)( xi µ x) =, (-5) ' ' E ( xi µ x)( yi µ y) E ( xi µ x)( xi µ x) ' ' ΛΓ ( xx Γ+ΨΛ+Σ ) εε = ' ' xx ΛΓ ΓΛ xx xx is the population covariance matrix of xs, and the meaning of the other symbols remain the same meanings as in the description of the unconditional model. There are two ways to incorporate time invariant predictors. One way, as described above, is to let the predictor impose direct effect on latent curve factors but only has indirect effect on outcome variables. This model is also called growth predictor model by Stoel, R.D., Van den Wittenboer, D. & Hox, J. (004). This is a widely used model in social science research. Among the 67 peer reviewed journal articles found by using key word latent growth searching in databases of Academic Search Premier, Business Source Premier, EconLit, Professional Development Collection, PsycINFO, and Sociological Collection from 004 to 008 more than 30% of studies employed this model. Stoel, R.D., et al. (004) argued that although this model had the distinctive advantage that the effect of time invariant covariate on growth parameters could be captured directly, the appropriateness of this model was based on the assumption of full mediation. That is, the direct effect of time invariant predictor on the outcome variable is equal to zero. If this assumption does not hold, the model is considered incorrect. Based on this argument, they proposed another way to incorporate time invariant predictors: regress predictors directly on outcome variables. This model was termed as direct effect model by Stoel, R.D., et al. (004). The model trajectory equation is described as follows: 30

31 y = α + λβ + γ x + ε, (-6) it i t i t i it where xi is the time invariant covariate for each individual and γ t is the regression coefficient between xi and y it.the subscript t for γ tindicates that the effect of x i on yit changes at different time. The level and shape equation is the same as equation -: α = µ + ζ i α αi B i = µ B + ζ Bi, (-7) where all the symbols remain the same meaning as before. Although this model is also widely used in applications, this study only focuses on growth predictor model. Latent growth model with a time-varying covariate The conditional model with a time varying covariate is more complex than model with a time invariant covariate in that the predictor varies with time (see Figure -3). The time varying covariate has to be added in the trajectory equation: y = α + λβ + γ x + ε, (-8) it i t i t it it where all the terms are the same as we specified in equation -6, except that x it is a time varying covariate measured for individual i at time t, and its effect on outcome variable yit is captured by coefficient γ t. The variable y it is now a function of level, shape, a time-specific influence of the covariate x it, plus a random error. The level and shape equation is the same as that in unconditional growth models: α = µ + ζ i α αi B i = µ B + ζ Bi, (-9) where all the symbols remain the same meaning as before. 31

32 σ 1 ζ α i μ α μ β ζ β i α i β i λ 1 λ λ t y i1 yi y it ε i1 γ 1 ε i γ ε it γ t x i i1 x i x it Figure -3. Latent growth model with a time varying covariate The structural equation form of this model is represented as follows: yi =Λ ηi +Γ χit + εi, (-30) where y i yi1 y i =, yi3 yi α i Λ=, η 1 i = β, i 1 3 γ1 γ, χ Γ= γ 3 γ 4 χi 1 χ ε i it = χi3 χi4 i εi1 ε i =. εi3 εi4 3

33 η = µ + ζ, (-31) i η i α i where ηi = β, i µ µ, ζ α η = µ B η ζ αi = ζ. βi According to this model, y is jointly affected by both the underlying random growth process and the time specific influences associated with the time varying covariate. A typical example of this model is the study conducted by Curran, Muthén and Hartford (1998), where they investigated time-specific impact of becoming married on heavy alcohol use. He tried to find out whether becoming married for the first time would affect heavy alcohol use controlling the normal development trend of alcohol use in early adulthood. This model is just appropriate for his research question. Latent growth model with a parallel process The previous two sections introduced two kinds of conditional LGMs that are also considered univariate LGM. That is, although there are multiple measurements on the outcome variable, they are multiple measures of one dependent variable. Sometimes we are interested in the analysis on more than one outcome variable. Suppose we have a dataset reflecting students academic performance at school. We might be interested in not only the growth trend in both individual mathematics and reading achievement but also whether the individual concurrent changes in the two areas are mutually interrelated. This allows us to understand the change in several domains and how these domains relate to each other. When LGM includes the latent curve process on more than one outcome variable, this type of model is called multivariate LGM. In this paper, it is referred as LGM with a parallel process. An example of a parallel process model is presented in Figure

34 ζ i1 ζ i ζ it x x i1 i x it α ix λ μαx ζαxi μ β x ζ β xi β ix σ ζ ζ αxi βxi γ γ β 1 γ α 1 β 1 λ λ t α iy μ α y ζ αyi 1 σ ζ ζ ζ β yi μ β y β iy i i 1 λ αy βy λ λ t y y i1 i y it ε i1 ε i ε it Figure -4. Latent growth model with a parallel process 34

35 For a simple illustration, only two outcome variables were included. The model equations for the variable y and for the variable x are described as: and where yit = αiy + λβ t iy + εyit, (-3) xit = αix + λβ t ix + εxit, (-33) α = µ + γ α + ζ iy αy α1 ix αyi = µ + γ α + γ β + ζ B iy By β1 ix β ix Byi α = µ + ζ ix αx αxi B ix = µ Bx + ζ Bxi αiy and, (-34), (-35) Biy represent the level and shape factor respectively for the outcome variable y; parameters αix and Bix represent the level and shape factor respectively for the variable x; parameters µ α y and µ By are the mean level and mean slope respectively for the outcome variable y controlling all other terms in their separate equation; parameters µ α x and µ Bx are the mean level and mean slope respectively for the outcome variable x; parameters ζ α yi and ζ Byi are still the disturbance for the level α iy and shape B iy respectively and the parameters ζ α xi and ζ Bxi are the disturbances of level and shape for the level α ix and shape B ix respectively. The coefficient γ α1 indicates the effect of initial status of the x variable on the initial status of the y variable. If the γ α1 is positive, higher growth status of the x would anticipate higher growth status of the y variable, after controlling the impact of the growth shape of the x variable. Parameter γ β1 captures the relationship between the level of the x variable and the shape of the y variable when the β is controlled. Parameter γ β represents the effect of growth shape of the x variable on the ix 35

36 growth shape of the y variable controlling the impact of α. A positive value of γ β1 indicates that high growth status of the x variable would predict faster growth on the y variable. A positive value of γ β would indicate that individuals growing quickly on the x variable would also tend to grow quickly on the y variable. One difference between the univariate LGM and multivariate LGM is that the latent factors have to be subscripted with y or x to differentiate the repeated measure of interest. With two outcome variables, the relationship between latent factors of one variable and the other one becomes much more complex. A point that is worthwhile to mention here is that there is no impact of the growth shape of the x variable on the level of the y variable. The rationale is obvious: the growth shape of the x variable is obtained later than the level of the y variable. Therefore, a future estimated variable can not be used to predict the current variable. The structural form of the model could be represented as follows: yi =Λ ηiy + εi, (-36) ix where y i yi1 y i =, yi3 yi Λ=, η 1 iy 1 3 εi1 α iy ε i =, εi =, and β iy εi3 εi4 η = µ +Γ ξ + ζ, (-37) iy η y i where η iy α iy =, β iy µ η y µ α y =, µ β y γ Γ= γ γ α1 α γ β1 β α ix, ξ = β, ζ ix ζ, and α η = ζ B χ =Λ ξ + δ, (-38) i i 36

37 where χi1 χ i χi =, χi3 χi4 1 0 δi1 1 1 Λ= α ix δ i, ξ = 1 β, δi =. ix δi3 1 3 δ i4 There are several variations of the parallel process model. In the model described above, only the level and shape of the y variable are predicted by the level and shape of x variable, not vice versa. In many studies, the level and shape of the x and y variables were predicted by each other in a variety of combinations. For example, the shape of the x variable can be predicted by the level of the outcome variable (e.g. Cheong, Mackinnon & Khoo, 003; Curran, 000). Therefore, the meaning of the outcome variable and predictor variable get blurred here. The key concept is that different domains are interrelated and are not independent of each other. All the variables must be assessed in the same measurement occasions. As pointed out by Muthén (00), one advantage of growth modeling in a latent variable framework was the ease with which to carry out analysis of multiple processes, both parallel in time and sequential. A variety of applications of this model have been discussed recently (e.g. Hudson, 008; Simons, 007; Mitchell, Kaufman, & Beals, 005). Assumptions of Growth Modeling The assumptions of growth modeling can be summarized in three aspects: within-person residual covariance structure, measurement time and missing data, and functional form of growth. Within-person residual covariance structure When the outcome variable is continuous, it is commonly assumed that the within-person error ε it is multivariate normally distributed with mean of zero and covariance matrix Θ ε. If the outcome variable is categorical, alternative estimation method would be used, such as weighted 37

38 least squares with corrected means and variance (Muthén & Khoo, 1998). Under the condition of categorical outcome variable, the assumption of multivariate normality should be relaxed. In a fashion analogous to the assumption in regression analysis, all the variables in the right hand side of the trajectory equation are uncorrelated with the error. More formally, take equation -1 as an example, that is, cov( εit, α i ) = 0 and cov( εit, β i ) = 0 for all i and t. The variance ofε it could be constant or non constant, depending on the data characteristic and real situation. Although it is mentioned in the introduction part that LGM allows the measurement error to be correlated across different time, it is not a general assumption. Many studies assume that the errors are not correlated over time, i.e., cov( ε, ε, + ) = 0 for s 0. It is also assumed that the it i t s errors of different individuals at different time are uncorrelated, that is cov( ε ε, + ) = 0 for i j and for s 0. When the errors are assumed to be uncorrelated over time, the assumption about the residuals is expressed as the follows: it j t s εi1 0 σe1 0 0 ε i 0 0 σe 0 N. (-39) εit σ et Regarding the level and shape equation, the unconditional LGM was used for a simple illustration: α = μ + ζ i α αi B i = μ B + ζ Bi. (-40) The disturbances ζ αi and ζ Bi are normally distributed with mean of zero and variance of αi and Bi. They are also correlated with each other with covariance. Furthermore, the two disturbances are assumed to be uncorrelated with the errorε it. 38